My campaign is still 1E/2E, so it's basically whatever the book says, rules-wise.
Culturally, half-elf PCs always have human mothers, and usually don't know anything about their fathers, because the PC is a changeling (in the "medievel folklore" sense, not the "shape-shifter or lycanthrope" sense) -- an elf-child that sneaky elves left behind after stealing a human baby. Or at least, that what's the human mothers usually say -- the elves say there are just too many human women who don't want to admit to a dalliance with an elf.
Half-elf babies look human, but become more elf-like as they mature. They usually don't have much standing in elven culture, because the elves are matrilineal, meaning the child of a human female doesn't have any significant relatives beyond fathers and siblings (even if they figure out who those are).
I know that might sound like a harsh background, but I wanted to incorporate folkloric changelings in my setting, and this was the least icky method I came up with. Now, for the totallly icky half-orcs:
Half-orcs are common-enough-to-be-PCs because the orcs in my campaign have adopted a viking-like "raid and run" lifestyle (with longboats, even), leaving a fair number of despoiled human (and goblin) women in their wake. Some of the resulting half-orcs (known as "harrowblooded" in the campaign world) actually make it to adulthood, if they don't get "lost in the woods" (sold into slavery, or just killed) by their parents.
Of course, there are a fair number of half-orcs living in the orc "kingdoms", too, where the orcs harass, dominate, and conquer other humanoids, so there are orc-goblins, orc-kobolds, and such running about, using stats from an old article in Dragon. On a strictly practical note, it's a great explanation for miniatures from different companies not quite matching -- "Oh, those weird kobolds? They're half-orcish!"
Yeah, life sucks if you're a half-orc. What else is new?
Half-elves and half-orcs aren't sterile, and can breed with any species their non-human parent can. Breeding "true" isn't breeding "simple" though, so it gets complicated after that.
Other than the possibilities of a quarter-elves and eighth-orcs, there are no weird or new PC half-breeds in my campaign. (I think we had a half-ogre once.) I have trouble envisioning a world where interspecies sex is so common as to support the level of cross-breeding seen in most 3E games. (Seriously, sometimes it's like the Forgotten Realms has turned into one big interspecies orgy.) As far as the NPC species go, I've kept the Fiend Folio troll half-breeds out of the game, because they're too weird for me.
If players decided to force my hand with odd interspecies relationships. I'd have to think about it. I know gully dwarves are supposed to be the result of gnomes and dwarves interbreeding, so I'd probably go that direction for a dwarf/gnome crossbreed. (Come to think of it, if the characters know that dwarf/gnome crossbreeding doesn't end well, it could explain an in-game taboo against the most obvious species-mixing.) I might left halfings reproduce with other species, only because I think Tolkien did, and halfings are the one species that's unavoidably tolkieneque in my campaign -- I can't even get players to stop calling them hobbits.
